To determine which of the given equations represents the line passing through the points [tex]\((-8, 11)\)[/tex] and [tex]\(\left(4, \frac{7}{2}\right)\)[/tex], we can follow the steps to find the equation of the line in the form [tex]\(y = mx + b\)[/tex]:
1. Determine the slope (m) of the line:
The slope formula [tex]\(m\)[/tex] for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points [tex]\((x_1, y_1) = (-8, 11)\)[/tex] and [tex]\((x_2, y_2) = \left(4, \frac{7}{2}\right)\)[/tex], we have:
[tex]\[
m = \frac{\frac{7}{2} - 11}{4 - (-8)}
\][/tex]
Simplify the numerator and denominator:
[tex]\[
y_2 - y_1 = \frac{7}{2} - 11 = \frac{7}{2} - \frac{22}{2} = \frac{7 - 22}{2} = \frac{-15}{2}
\][/tex]
[tex]\[
x_2 - x_1 = 4 - (-8) = 4 + 8 = 12
\][/tex]
Thus, the slope [tex]\(m\)[/tex] is:
[tex]\[
m = \frac{\frac{-15}{2}}{12} = \frac{-15}{2} \cdot \frac{1}{12} = \frac{-15}{24} = \frac{-5}{8}
\][/tex]
2. Find the y-intercept (b) using one of the points:
Use the point-slope form of the equation [tex]\(y - y_1 = m(x - x_1)\)[/tex] to find the y-intercept. For simplicity, we can use point [tex]\((-8, 11)\)[/tex]:
[tex]\[
y = mx + b
\][/tex]
Substituting [tex]\(m = -\frac{5}{8}\)[/tex] and [tex]\((x, y) = (-8, 11)\)[/tex] into the equation:
[tex]\[
11 = -\frac{5}{8}(-8) + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
11 = \frac{40}{8} + b
\][/tex]
[tex]\[
11 = 5 + b
\][/tex]
[tex]\[
b = 11 - 5 = 6
\][/tex]
Therefore, the slope-intercept form of the line passing through the points is:
[tex]\[
y = -\frac{5}{8}x + 6
\][/tex]
3. Match the derived equation with the given options:
The provided options are:
- [tex]\(y = -\frac{5}{8}x + 6\)[/tex]
- [tex]\(y = -\frac{5}{8}x + 16\)[/tex]
- [tex]\(y = -\frac{15}{2}x - 49\)[/tex]
- [tex]\(y = -\frac{15}{2}x + 71\)[/tex]
The equation [tex]\(y = -\frac{5}{8}x + 6\)[/tex] is the correct match.
Thus, the equation that represents the line passing through [tex]\((-8, 11)\)[/tex] and [tex]\(\left(4, \frac{7}{2}\right)\)[/tex] is:
[tex]\[
y = -\frac{5}{8}x + 6
\][/tex]
